In early colonial days there were both poverty and hard work, the work being of such a nature that cuts, bruises, and blows could well have been the daily fare of our forebears.
When the injury was to the shin, it was usual to apply a poultice of sorts, and, among the poorer classes, this often took the form of a small square of paper previously soaked in vinegar, tobacco juice, or some other decoction of soothing, if not medicinal, value.
These, of course, were shinplasters.
With the advent of paper currency of dubious value, the size thereof being reminiscent of the makeshift poultice and the value often also of the same magnitude, the term was transferred to the scrip, and even yet paper scrip of low real or imagined value is known by the same term.